Frenet-Serret and Darboux Frames of a Image Curve on a Screw Surface

Sinop Üniversitesi Fen Bilimleri Dergisi Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

Research Article

Frenet-Serret and Darboux Frames of A Image Curve on A Screw Surface

Yasemin Kemer *a, Erhan Ata a

a Dumlupınar University, Faculty of Science and Arts, Department of Mathematics, Kütahya, Turkey

Abstract

In literature, the action of the dual quaternion which described a rigid motion on a point and a line were investigated. Based on this idea, one can obtain a new surface by applying the rigid motion to the points of a surface. In this paper, we generate a screw surface M v by applying the screw motion, a special case of the rigid motion, to the points of a surface M in E3 . In , we obtain the frames which correspond to Serret-Frenet and Darboux frames of a curve  on M . These frames are Serret-Frenet and Darboux frames of the image curve on . Then, we calculate the curvature, torsion, geodesic curvature, normal curvature and geodesic torsion of the image curve of the curve  on the screw surface M v and obtain the relationships between these notions of the curve  and its image curve.

Key Words: Curvature, Darboux frame, Frenet frame, image curve, rigid transformation, screw surface, torsion.

Vida Yüzeyi Üzerindeki Bir Görüntü Eğrisinin Frenet-Serret ve Darboux Çatıları

Öz

Literatürde katı cisim hareketini tanımlayan dual kuaterniyonun nokta ve doğru üzerindeki etkileri incelenmiştir. Buradan hareketle bir yüzeyin noktalarına katı cisim hareketi uygulanarak yeni bir yüzey elde edilebilir. Bu çalışmada, 3-boyutlu Öklid uzayında bir yüzeyinin noktalarına katı cisim hareketinin özel hali olan vida hareketini uygulayarak bu yüzeye ait bir vida yüzeyi oluşturulmuştur. yüzeyi üzerindeki bir eğrisinin Serret- Frenet ve Darboux çatılarına vida yüzeyi üzerinde karşılık gelen çatılar elde edilmiştir. Bu çatılar eğrisinin vida yüzeyi üzerindeki görüntü eğrisine ait Serret-Frenet ve Darboux çatılarıdır. Daha sonra, görüntü eğrisine ait eğrilik, burulma, geodezik eğrilik, normal eğrilik ve geodezik burulma hesaplanıp eğrisi ve bu eğriye ait görüntü eğrisinin hesaplanan bu kavramları arasındaki bağıntılar elde edilmiştir.

Anahtar Kelimeler: Burulma, Darboux çatısı, eğrilik, Frenet çatısı, görüntü eğrisi, katı cisim hareketi, vida yüzeyi.

* Corresponding author Received: 19.10.2016 e-mail: [email protected] Accepted: 21.02.2017

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

Introduction properties of the curve independent from movement. This formulae was named from Surfaces have had application areas two French mathematicians Jean Frederich in many areas such as mathematics, Frenet and Joseph Alfred Serret who kinematics, dynamics and engineering for discovered this formulae independent from many years and they have been in center of each other. interest increasingly. Mathematicians have written many articles and books by Darboux frame is a nature moving investigating surfaces as Euclidean and frame setting up on a surface in differential non-Euclidean. For these studies, one can geometry of surfaces. This frame was read [1], [2], [3], [4], [5], [6], [7], [8]. named from French mathematician Jean Gaston Darboux. Darboux frame, as Eisenhart defined parallel surfaces Frenet-Serret frame, is an example of a and their some properties in his book [3]. nature moving frame defined on a surface. In [9], Ünlütürk and Özüsağlam In this study, we obtain the image investigated the parallel surfaces in curve of a curve  on a surface M in Minkowski 3-space. In [10], Tarakçı and Euclidean 3-space E3 on a screw surface Hacısalihoğlu defined surfaces at a M v . Then we calculate Serret-Frenet and constant distance from edge of regression Darboux frames of the image curve and we on a surface and gave some properties of get the curvature, torsion, geodesic such surfaces and then in [11], [12], [13] curvature, normal curvature and geodesic Sağlam and Kalkan investigated the other torsion of this curve. These notions are properties of this surface. Again Sağlam expressed in terms of the corresponding and Kalkan transported the surfaces at a values of the the curve  . In this way, we constant distance from edge of regression investigate the changes in some geometric on a surface to Minkowski 3-space and properties of a curve under the screw obtained their properties which they have motion. This paper depends on the case in Euclidean space in this space. that position vector of points of a surface In her Ph.D. thesis Kemer described and the unit normal vectors at these points points of a surface M in Euclidean 3- are parallel. space E3 by dual quaternions and applied screw motion, which is a special case of Preliminaries rigid motion, to these points and obtained a screw surface M v [14]. The screw Quaternions surfaces are a more general case of the parallel surfaces and surfaces at a constant Let us firstly begin with Hamilton's distance from edge of regression on a quaternions and their connection with surface. In special cases, one can obtain the rotations. A rotation of angle  about a T parallel surfaces and surfaces at a constant unit vector v = (vx ,vy ,vz ) is represented distance from edge of regression on a by the quaternion, surface from the screw surfaces.   In differential geometry, Serret- r = cos  sin (vxi  vy j  vzk). 2 2 Frenet formulae define the kinematic The conjugation properties of an object moving along a p = rpr continuous, differentiable curve in gives the action of such a quaternion on a Euclidean 3-space E3 or geometric point p = xi  yj  zk in space, where

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

  be an arbitrary speed curve on this surface. r = cos  sin (v i  v j  v k). 2 2 x y z Then we call the vectors

The quaternions representing ()t Tt( ) = , rotations satisfy rr = 1 and also r and r ()t represent the same rotation. The set of unit quaternions, those satisfying rr = 1, N()()() t B t T t comprise the group Spin(3), which is the and double cover of the group of rotations ()()tt SO( 3) . Bt()= ()()tt Let  be the dual unit which as the unit tangent, the unit normal and the satisfies the relation  2 = 0 and commutes binormal vectors of the curve  , with the quaternion units i, j and k. For respectively. The triple {T , , N} B defines a ordinary quaternions q and q , 0 1 frame in E3 and this frame is called Frenet h = q0  q1 frame [16]. indicates a general dual quaternion. A rigid transformation is represented by a dual Let U be the unit normal vector of quaternion M . Then the triple {T , , V} U defines a 1 g = r  tr, frame in E3 and this frame is called 2 where r is a quaternion representing a Darboux frame, where VUT=  [17]. rotation as above and t = t i  t j  t k is a x y z Theorem 1. pure quaternion representing the translational part of the transformation Let M be a surface in E3 and  be an [15]. arbitrary speed curve on this surface. Then the curvature function  (t) can be Points in space are represented by dual quaternions of the form, calculated as ()()tt pˆ = 1 p, ()t = 3 where p is a pure quaternion as above. The ()t action of a rigid transformation on a point [3]. is given by, Let M be a surface in E3 and 1 1 pˆ'= (r  tr) pˆ(r  rt) be the Darboux frame of an 2 2 arbitrary speed curve  on this surface. 1 1 = (r  tr)(1  p)(r  rt) Let v = (t) . The functions 2 2 = 1 (rpr  t). 1 kg ( t ) =2  ( t ), V ( t ) , Note that, as with the pure rotations, v g and g represent the same rigid 1 kttn ( ) U =( t ),(2  ) , transformation [15]. v 1  (t ) = V ( t ), U ( t ) Darboux and Frenet Frames in E3 g v are called the geodesic curvature, the Let M be a surface in E3 and (t) normal curvature and the geodesic torsion

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383 of the curve , respectively [18]. A curve M in E3 . Let the vector fields v for which kn = 0, kg = 0 or  g = 0 at X  (M ) , X  (M ) be given by the every point on the surface is called an property that for p  M , asymptotic curve, a geodesic curve or a b p( b) = f p ( ( ) ) , 13i with respect to line of curvature, respectively [19]. ii the Euclidean coordinate system x1, x2, x3 Screw Surfaces of the space E 3, where

v 3 Let M and M be two surfaces in E and 3  3  p  M . Let X = bi , X = bi . i=1 xi i=1 xi  Then rvvv= cossin() ijk xyz 2   2   22 1. fX p = cos X p   p sin   S(X p ) be a rotation by an angle of  radian about 2  2  v 2. S ( f(X )) f ( p) = S(X ) p . the unit vector v= ( vx , v y , v z ) and tv=  be the translational vector. If there is a [14]. function defined as v Proof. f : M  M p  f ( p) = rpr  t 1. The points of the screw surface M v of then the surface M v is called a screw a surface M are obtained by the function surface of the surface M [14]. Let the f : M  M v as rotation axis be the unit normal of the surface M at the point p , i.e. let vU= .

In this case the function of a screw surface is The vectors on the surface M can be  v f( ppUpU ) = cossin2sin,. p UU 2 transferred to the surface M by the map 2 a If the position vector of the point p fIp cossin22 i p 3  22x j and the unit normal vector U are parallel p 1  then this equation becomes as  sin2 pa . p 2 ij 22 3  fpppU() =. cossin  p where Ua= is the unit normal (1) 22  i i=1 xi vector field of the surface M . Then for This paper consists of this case. All X T M we have operations will be done according to this p p case in the rest of the study. 2   2   f X p = cos X p   p sin   S(X p ). p 2  2  Following theorem shows how to transfer tangent vectors of a surface to the 2. Let and S be the unit screw surface of this surface: normal vector field and shape operator of Theorem 2. the surface M , respectively. Let us represent the unit normal vector field of v Let M be the screw surface of a surface M v by U v and the shape operator by S v .

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

Then at the point f ( p) the normal vector Darboux frame of this curve is of M v is  33 vfT   v v v v v  TVUTU= , = , UafpapUfpp()=()==,     iixx v ii=1=1 iipp  where v f= T . where ai  f = ai .   

3  Following theorem shows how to Assume that Xbp=   is the unit v pi x obtain Darboux frame of the surface M at i=1 i p the point f (p) = f (t). tangent vector of a curve  : I  M at the point p  M . In this case the tangent Theorem 3. v vector fX p Tf ( p)M is also the tangent vector of the curve f  : I  M g at the Let T V,, U  be the Darboux frame point f ( p) M g . Thus, at the point of a curve  on M . Then the Darboux p = (t) we have frame of the image curve  =  f  on 3  3 d(a ) M v at the point f (p) = f (t) is S(X ) = X a  = i p   (t) i x  dt i=1 i  (t) i=1  (t) 1  3 v 22 daf() Tfp()= cos knp p sin  T = i v 22  dt i=1  ()t

2   3  gppVsin   f X  a . 2    fp()i x i=1 i p v 1 22 VkpVfp()= cossin np v 22 On the other hand, since 

3   v v  pT2 S ( f(X )) f ( p) = D f ( X ) U = X p ai  gpsin   f ( p)  x 2 i=1 i p  v 3  UUfpp()= = f X a    f ( p)  i  x where kn and  g are normal curvature and i=1 i f ( p) geodesic torsion of M , respectively. we get Sv ( f (X )) = S(X ) .  f ( p) p Proof.

Darboux Frame of A Image Curve on Since the derivative of the unit normal of Screw Surface M at the point p = (t) is

v  Let M be the screw surface of a Up=  k n T p g V p , the tangent vector of surface M and  =  f  be the image the curve  at the point f ( p) = f ((t)) is of a unit speed curve  on M . Then, (f ( p )) = f T  '= fT   . Since the curve  is not a   p  unit speed curve at the point ft() ,

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

v 22 screw surface M . Let  be a unit speed = cossin kpTnp 22 curve on M ,  =  f  be the image of  on the screw surface M v and 2  gppVsin . 2 be the Darboux frame of Then we have the curve  at the point f ((t))  M v . v= f T    Then  2 vv'vv'v 22 kftTUUTn ((( ))) =,=,  =  coskpn  sin  ftft(( ))(( )) 22  is called the normal curvature of the screw 2 1/2 surface M v . Let  be a unit speed curve 2   p 2 . g sin  on M ,  =  f  be the image of  on 2  the screw surface M v and Therefore we get be the Darboux frame of

the curve  at the point f ((t))  M v . v 1 22 TkpTfp()= cossin np Then v 22 vv'vv'v g ((ftVUUV ( ))) =,=,  ftft( ( ))( ( )) 2   is called the geodesic torsion of the screw gppVsin . 2  surface M v .

Since M v is the screw surface of M we Theorem 4. v have UUf() p= p . Then Let  be a unit speed curve on M and  = f  be the image of  on the vvv   VUTfp()=    v fp() screw surface M . Then geodesic curvature, normal curvature and geodesic torsion of  =  f  at the point f ((t)) 1 22 = cossin kpVnp are v 22

2 2     pT. v'1  gpsin  k=  k p 2  2  g3 g n sin v   2

Let  be a unit speed curve on M , 22 2 cossinpk v  gg  =  f  be the image of  on the 22 v TVUvvv,, pT, 2  screw surface M and p 222 '   ft( ( ))  kp  sin cossin n  be the Darboux frame of the curve  at p 222  v the point f ((t))  M . Then v 1  2  2 2  2   kn = 2 kn cos  kn  g  p sin    v  2  2  v v' v v' v kg ( f ( ( t ))) = T , V = V , T and f( ( t )) f ( ( t ))   is called the geodesic curvature of the  v = g cos2 , g v2 2

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

respectively. vv1 kUn =,2  v fp() Proof.

Since 1 2222 =.2 kkpnngcossin   v 22   22Finally, one can get the geodesic torsion as (())fpkpT = cossin np 22 vvvv 11  g =',=',VUUV  vvfpfp()()

2  gppVsin 2  g 2   = c o s . we have v2 2  pT, p (())fpkpk =  sinsin22  ggn   22p Corollary 1.    2 kpTnpsin   2  The image curve of a geodesic curve is also geodesic if and only if the curve on   222 the main surface M is a line of curvature. knncos  k p sin    22 Proof. 2 2   gppUsin  2  If the curve on the main surface M is a  geodesic then the geodesic curvature of 22  kkgncossin kp gg ()  22this curve is kg = 0 . If this curve is a line

pT,  of curvature then we have  g = 0. Hence, p 2   sin V . v gp we get k = 0 and this completes the p 2  g proof. Since  =  f  is not a unit speed curve, Corollary 2. the geodesic curvature of M v is The image curve of an asymptotic curve is vv1 also asymptotic if and only if the curve on kVg =,2  v fp() the main surface M is a line of curvature.  1  pT, p  =   sincos22kv2 Proof. 3 gg vp  22  If the curve on the main surface M is an asymptotic curve then the normal curvature 22  22of this curve is kn = 0. If this curve is a kpkpng n sinsin  22   line of curvature then we have  g = 0. v 22 Hence, we get kn = 0 and this completes cosp sin  . 22 the proof.

The normal curvature can be obtained as Corollary 3. The image curve of a line of curvature is

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383 also a line of curvature. Therefore, we can give the following theorem: Proof. Theorem 5. Since the curve on the main surface M is a line of curvature, then we have  g = 0. Let  =  f  be the image of  on the v v Therefore, we can obtain  g = 0 and this screw surface M and be the completes the proof. Darboux frame of  on M . Then the Frenet frame of  =  f  at the point Frenet Frame of A Image Curve on v Screw Surface f ( p) = f ((t)) on the screw surface M is Let  =  f  be the image of  v 1  on the screw surface M and TVU,, be v 22   Tfp()= cos knp p sin  T v 22 the Darboux frame of  on M . Since the curve  is not a unit speed curve, we have 2   gppVsin .  2   2 v 2   |=f() pn gp v kpT sin 2 vv1  2  NkpT|=f() pg gp  sin  vv22 32vv22  2 vkk ng v kgpnnp Uv kkpV cossin  .     22  It follows that 3 vv22 =.vkk  ng   fp() Therefore, we get

  vv1  2  v  BkpT|==  B |=fp() f() pn gp  sin   vv22 2 vkk ng   

1  v  2 = kn g psin T p vv22 2 v kng   k 

vv22 vkgpnnp UkkpV cossin  . 22 On the other hand, one can obtain Theorem 6. that vvv v v v Let TNB,, be the Frenet frame of NBT|=    fp()   fp() the curve  =  f  on the screw surface 1  v 2  v = kpTg gpsin  and  be the angle between the vv22 2   v v kkng    normal N of the curve  and the unit

v v normal U of the surface M . Then the vv22 curvature of  is vkn U p  k gcos  k n p sin   V p . 22

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

v v 2 v 2 them two units in direction of . After  = kn   kg  and the necessary calculations we get the screw surface k v c o sv = n . Mv = 2cos u cos v ,  2cos u sin v ,  2sin u | K v 

Proof. uv0,0 22 Since of the surface M . Let us consider the curve   Cucosusinuu=(,0) = (0,,) |0,  v  2  = 3   v  0. 2  we get 2 2 v3 k v   k v  Darboux frame of this curve at the point  v = n g p = (0,1,0) M is v3 0

2 2 Tp( ) = ( 0 ,0 ,1 ) , = k v   k v  . 0 n g Vp( ) = ( 1 ,0 ,0 ) , On the other hand, 0 Up() = (0,1,0). cos=,vvv NU 0 One can easily calculate the

1 vvv geodesic curvature, the normal curvature =,vkUUn and the geodesic torsion of the curve at the vv22 vkk ng    point p0 as kpg (0 ) = 0, kpn ()=10 and k v  ( p ) = 0 , respectively. Frenet frame of = n . g 0 K v the curve C can be obtained as Tp() = (0,0,1), Example. 0

Np(0 ) = (0,1,0), Consider the surface Bp() =0 (1,0,0). M= ( u , v ) = cos u cos v ,cos u sin v ,sin u |

uv0,0 . 22 The unit normal of the surface is U= cos u cos v ,  cos u sin v ,  sin u .

Let the rotation axis be U p , rotation angle be  and the translation vector be 2U 2 p for p = (u,v) M . Then the screw motion rotates the points of M about by an angle of radian and translates Figure 1: Serret-Frenet Frame of A Image Curve

Kemer and Ata Sinop Uni J Nat Sci 2(1): 48-58 (2017) ISSN: 2536-4383

On the other hand, image of the point References

p0 on the screw surface will be the point

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